HK-PC 2023

Let $ABCD$ be a convex quadrilateral with $AB=5$ , $AD=17$ , and $CD=6$ . If the angle bisector of $\angle{BAD}$ and $\angle{ADC}$ intersect at the midpoint of $BC$ , find the area of $ABCD$ .

(a) Find the smallest number of lines drawn on the plane so that they produce exactly 2022 points of intersection. (Note: For 1 point of intersection, the minimum is 2; for 2 points, minimum is 3; for 3 points, minimum is 3; for 4 points, minimum is 4; for 5 points, the minimum is 4, etc.) (b) What happens if the lines produce exactly 2023 intersections?